Sanchez has scored 13 goals in 24 appearances for Arsenal since moving to the Emirates Stadium from Barcelona in a deal worth a reported £35million.
Yet Rodgers, whose side host Arsenal on Sunday, was eager to bring the Chile international to Anfield as a replacement for Luis Suarez.
Liverpool have struggled for goals this season, with the absence of Daniel Sturridge due to a string of injuries coming at a particularly inconvenient time following Suarez's departure to Barcelona.
Discussing his club's attempt to bring in Sanchez, Rodgers said: "All I know is that he's a world class player. He was identified as someone who could come in and be perfect for us.
"He's a brilliant player with outstanding quality and even bigger work-rate.
"I think if you look at how I ask teams to play speed is critical. Not just with the ball, but the intensity with the pressing.
"When Daniel Sturridge was injured in August I think everyone who knows football should hopefully be able to see that our game has become much deeper because of personnel.
"Sanchez was identified for us as someone who would have been a key signing and really just a roll on to what we had with Luis Suarez.
"So to not get him was obviously bitterly disappointing, but once he was gone that was it. We just had to focus on what we had.
"He would have been perfect for us. He made his decision. The best option for him."
Liverpool head into Sunday's meeting with Arsenal trailing the London club, who sit sixth ahead of the weekend fixtures, by five points.
